JHEP03(2018)177

Published for SISSA by Springer

Received: January 29, 2018

Accepted: March 2, 2018

Published: March 28, 2018

Cosmic censorship at large D: stability analysis in

polarized AdS black branes (holes)

Norihiro Iizuka,a Akihiro Ishibashib and Kengo Maedac

aDepartment of Physics, Osaka University,

Toyonaka, Osaka, 560-0043 JapanbDepartment of Physics, Kindai University,

Higashi-Osaka, Osaka, 577-8502 JapancFaculty of Engineering, Shibaura Institute of Technology,

Saitama, 330-8570 Japan

E-mail: iizuka@phys.sci.osaka-u.ac.jp, akihiro@phys.kindai.ac.jp,

maeda302@sic.shibaura-it.ac.jp

Abstract: We test the cosmic censorship conjecture for a class of polarized AdS black

branes (holes) in the Einstein-Maxwell theory at large number of dimensions D. We first

derive a new set of effective equations describing the dynamics of the polarized black

branes (holes) to leading order in the 1/D expansion. In the case of black branes, we

construct ‘mushroom-type’ static solutions from the effective equations, where a spherical

horizon is connected with an asymptotic planar horizon through a ‘neck’ which is locally

black-string shape. We argue that this neck part (of black string) cannot be pinched off

dynamically from the perspective of thermodynamical stability. In the case of black holes,

we show that the equatorial plane on the spherical horizon cannot be sufficiently squashed

unless the specific heat is positive. We also discuss that the solutions are stable against

linear perturbation, agreeing with the thermodynamical argument. These results suggest

that Gregory-Laflamme type instability does not occur at the neck, in favor of the cosmic

censorship.

Keywords: Black Holes, Classical Theories of Gravity

ArXiv ePrint: 1801.07268

Open Access, c© The Authors.

Article funded by SCOAP3.https://doi.org/10.1007/JHEP03(2018)177

JHEP03(2018)177

Contents

1 Introduction 1

2 Effective equations in charged AdS black branes 2

3 Black mushroom solutions 5

4 Stability analysis 8

4.1 Stability analysis: thermodynamical argument 9

4.2 Stability analysis: linear perturbation 10

5 Spherical black hole case 12

5.1 Derivation of effective equations 12

5.2 Properties of static solutions 13

5.3 Linear perturbations 15

6 Summary and discussions 16

A Formula for curvature decomposition 16

1 Introduction

In contrast to asymptotically flat spacetimes, there is a large variety of asymptotically

Anti de Sitter (AdS) black hole solutions due to the warping factor. For instance, given

an asymptotically AdS charged black hole, one can deform it by applying a non-uniform

electric field and thereby construct a new black hole solution without destroying the asymp-

totic AdS structure, as demonstrated by [1]. Related to this, the possibility of AdS solitons

with electric multipoles was suggested by [24]. This fact leads to the recent discovery of

four-dimensional polarized AdS black holes in a dipolar electric field [2]. (See also [25] for

an explicit construction of AdS black holes with higher electric multipoles, and [26] for po-

larized black holes in different gravity theories.) The solution was numerically constructed

as a generalization of the Ernst solution [3]. Such polarized AdS black holes were extended

into four-dimensional polarized AdS black brane solutions with a planar horizon, where

chemical potential varies along one spatial direction [4]. In the latter black brane case,

by locally applying sufficiently large enough and localized chemical potential, it is shown

numerically that the configuration of the horizon looks like a mushroom and hence is called

a “black mushroom” solution [4].

In this black mushroom solution, a neck connecting a localized spherical black hole and

asymptotic planar horizon appears. The neck of the black mushroom solution is getting

thiner as the temperature is lowered, and is expected to behave like a thin black string. This

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JHEP03(2018)177

immediately leads us to the question of whether thin neck part is pinched off dynamically

due to the Gregory-Laflamme instability [5]. If that is the case, a naked singularity would

appear and the cosmic censorship [6] would be violated in the polarized black brane. There

have also been a number of numerical study for the violation of the cosmic censorship in

higher dimensions. The goal of this paper is to supply an analytic study for the cosmic

censorship conjecture in the class of polarized AdS black branes (holes) in the Einstein-

Maxwell theory, namely, mushroom-type and other type of black branes (holes).

In order to attack this issue analytically, we adopt the large D effective theory approach

developed in refs. [7–14]. We first derive a tractable set of effective 1 + 1-dimensional

equations describing the dynamics of deformed charged AdS black branes (holes) in the

leading order of the 1/D expansion. Then using these effective equations, as in ref. [4],

we obtain polarized black mushroom solutions with a neck connecting a localized spherical

horizon and an asymptotic planar horizon. Near the neck, the horizon geometry locally

behaves as a black string, and it is polarized by strong electric field along the neck. Applying

the claim of the Gubser-Mitra conjecture [15, 16], which has now been proven for some

cases [17], to the local black string, we argue that the neck should be locally stable against

physically reasonable perturbations, conforming to the thermodynamic stability.

We also find polarized AdS black hole solutions in which the spherical horizon is

squashed around the equatorial surface. One may expect that there could be a black

“dumbbell” solution whose horizon looks like a dumbbell having two spherical horizons

connected by a portion of a thin black string. We show however that the equatorial plane

on the spherical horizon cannot be sufficiently squashed while keeping its local specific

heat negative to lead an instability. This implies that there is no black dumbbell solution,

where two spherical horizons are connected through a thermodynamically unstable thin

black-string-shape neck, and therefore the neck cannot be pinched off dynamically due to

the Gregory-Laflamme instability [5]. We also discuss that the solutions are stable against

linear perturbations, being consistent with the thermodynamical argument.

The organization of this paper is as follows; in the next section, we first derive the

1 + 1-dimensional effective equations by expanding the Einstein equations in the inverse

power of D. In sections 3 and 4, we construct the black mushroom solutions and test the

cosmic censorship conjecture in them. In section 5, we repeat the analysis in the polarized

AdS black hole solutions. Section 6 is devoted to summary and discussions.

2 Effective equations in charged AdS black branes

We start with the following D-dimensional Einstein-Maxwell equations with a negative

cosmological constant

Rµν −1

2Rgµν + Λ gµν =

1

2

(FµρFν

ρ − 1

4F 2gµν

),

Λ = −(D − 1)(D − 2)

2L2,

1√−g

∂µ(√−gFµν

)= 0, (2.1)

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JHEP03(2018)177

where L is the AdS curvature length and Fµν = ∂µAν − ∂νAµ. We make the following

ansatz for the metric and the gauge field as

ds2 = −Adt2 + 2ut dtdr − 2Cz dtdz +Gzzdz2 +

r2z2

L2dΩ2

n−2,

A = Atdt+Azdz, (2.2)

where dΩ2n−2 is the metric of unit sphere with n = D − 1. Note that we do not rescale z-

coordinate, as done in [18] since z is not the direction of Killing symmetry of the background

geometry.

For simplicity, we assume that at large D, the gauge field Aµ behaves as

At = O(n−

12

), Az = O

(n−

32

). (2.3)

Then, the electric charge of the black brane can be dealt with a test charge so that it does

not affect the metric at leading order in the expansion in the inverse of n [19]. This leads

to the metric expansion as follows

A(r, t, z) =r2

L2

(1− m(t, z)

rn

)+

r2

nL2

(Q(t, z)

r2n−2+ a1(r, t, z)

)+O(n−2),

Cz(r, t, z) =p(t, z)

nrn+O(n−2),

ut(r, t, z) = 1 +βt(r, t, z)

n+O(n−2),

Gzz =r2

L2+H(r, t, z)

n+O(n−2), (2.4)

where the horizon is determined by A = 0. It is convenient to use the formula (A.1) to

expand the Einstein eqs. (2.1) order by order as a series in 1/n. Then, we find that the

metric given above already solves the Einstein equations at leading order.

We would like to derive the effective equations for the variables, m(t, z), p(t, z), · · · .For that purpose, let us define R as R = (r/r0)n, where r0 is a fiducial horizon size.

Hereafter, without loss of generality, we set r0 = 1. We take the large D (or equivalently

large n) limit in such a way that R = fixed, i.e., r → 1, n → ∞ with rn = fixed. Note

that this limit forces us to set the finite power of r to be 1 in the leading order of large

n expansion. Within this limit, we evaluate the Einstein & Maxwell equations in the 1/n

expansion at the horizon and derive the effective equation for the variables. Note that the

horizon is determined as R = rn = m(t, z) in the leading order of large n expansion.

With this double scaling limit in our mind, as for the gauge field we make the ansatz

for At as

At(r, t, z) =

√2

n

(P (t, z)− q(t, z)

rn

). (2.5)

Here, P plays a role of the chemical potential on the AdS boundary, r =∞ and ∂zP cor-

responds to the external electric field along z-direction. Then the t-component of Maxwell

– 3 –

JHEP03(2018)177

equation at the leading order is automatically satisfied. The function q will correspond to

the electric charge as we will see below.

Substituting eqs. (2.4) and (2.5) into the z-component of the Maxwell equations in

eqs. (2.1), and by evaluating its leading order at the horizon in the 1/n expansion, we

obtain

− 1√2

∂A

∂r

∂Az∂r−√npq

r2n+√n∂P

∂z= 0 . (2.6)

This gives a solution for Az as

Az(r, t, z) =

√2

n

L2

n

(∂P (t, z)

∂zln(rn) +

p(t, z)q(t, z)

m(t, z)rn

). (2.7)

At next to leading order in 1/n expansion, from the rr, rz, zz-component of the

Einstein equations, we find that βt and H can be set to zero:

βt = H = 0 . (2.8)

From the several components of the Einstein equations (2.1), we obtain

Q = L2q2(t, z). (2.9)

Then, rt-component of the Einstein equations (2.1) reduces to

∂a1

∂r+

1

n

∂2a1

∂r2=nL4p

Rz, (2.10)

and its solution is given by

a1 = −L4p lnR

zR. (2.11)

From the tt and tz-component of the Einstein equation, the evolution equations for m and

p are obtained as

∂m

∂t+L2

zp− L2

z

∂m

∂z= 0 , (2.12)

∂p

∂t+L2

mzp2 − 2q

∂P

∂z+

1

L2

∂m

∂z− L2 ∂

∂z

(pz

)= 0 , (2.13)

on the horizon R = rn = m. Finally the r-component of the Maxwell equations yields the

evolution equation for q:

∂q

∂t− L2

z

∂q

∂z+L2m

z

∂P

∂z+L2p

zmq = 0 . (2.14)

These three equations (2.12), (2.13), and (2.14) are the 1+1-dimensional effective equations

for the charged black brane. As far as we are aware, these are completely new equations.

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JHEP03(2018)177

3 Black mushroom solutions

In this section, we derive a black mushroom solution from our effective equations (2.12),

(2.13), and (2.14). The topology of the horizon at t = constant surface is Rn−1 and the

metric becomes

ds2fixed t =

r2(z)

L2(dz2 + z2dΩ2

n−2) , (3.1)

where z is the radial coordinate on the horizon and r(z) is the location. In the black

mushroom solution there is a neck, as found in ref. [4]. By denoting the area of the

z = constant. surface as S(z), the minimum condition for the existence of a neck (located

at z = z0 (> 0)) can be geometrically defined as

∂S(z)

∂z

∣∣∣∣z=z0

= 0,∂2S(z)

∂z2

∣∣∣∣z=z0

> 0, m(z0) < m|(z→∞),

S(z) :=C0

LnR(z)zn =

C0

Lnm(z)zn, (3.2)

where C0 is the surface area of unit n− 2-dimensional sphere. The third condition on the

first line implies that there should be a concavity for the horizon radius RH in the range

0 < z < ∞. When m = constant., the solution becomes a plane-symmetric charged black

brane solution with no neck. As is shown below, the neck can be created only by the

localized chemical potential, P (z), or it would be more correct to say that the neck can be

created by the strong external electric field, ∂zP .

Hereafter, we set L = 1 for simplicity. Making the ansatz

m = m(z), p = p(z), q = q(z), P = P (z) , (3.3)

for the static solution, we reduce eqs. (2.12), (2.14), and (2.13) to

m′ = p ,

q′ =pq

m+mP ′ ,(p

z

)′= m′ +

p2

mz− 2qP ′ , (3.4)

where the prime means the derivative with respect to z. The horizon is determined by

A = 0 as

RH = m− a1m

n− Q

nm+O

(1

n2

). (3.5)

The surface gravity κ is given by

κ :=1

2

∂A

∂r

∣∣∣∣R=RH

=1

2

(n+ lnm− p

zm− q2

m2

)+O

(1

n

). (3.6)

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JHEP03(2018)177

In order to derive this, one has to be careful to the fact that

r = 1 +1

nlnm+O

(1

n2

). (3.7)

It is easily checked from eqs. (3.4) that κ is constant along the horizon by showing that

∂κ

∂z= O

(1

n

). (3.8)

Note that eq. (3.7) indicates that deformation from the homogeneous black brane solution

is O(1/n) in our black mushroom solution, while it is O(1) in the black mushroom solution

numerically constructed in ref. [4]. Nevertheless, as we will show, our black mushroom

solution has a neck defined in eq. (3.2).

Now, we will construct a black mushroom solution which is deformed by the external

electric field P ′. Substitution of p = m′ into the second equation in (3.4) yields

q′

m− m′

m2q = P ′. (3.9)

This can be integrated as

q

m= P + C, (3.10)

where C is an integral constant. As an asymptotic boundary condition at infinity, z →∞on the horizon, we will impose that the black brane solution asymptotically approaches a

uniformly charged black brane solution. This is equivalent to impose the following condi-

tions,

limz→∞

P =q0

m0, lim

z→∞m = m0 (> 0),

limz→∞

q = q0 (> 0) . (3.11)

The first condition implies that there is no external electric field at infinity. The boundary

condition determines the integral constant C as

C = 0 . (3.12)

This is consistent with the regularity condition on the horizon, At = 0 in eq. (2.5) (see, for

example, ref. [20]).

Introducing new variables M and ξ as

m = m0 eM , q = ξ eM , (3.13)

we obtain the equation of motion for M from the third equation in (3.4) as

M ′′

z−(

1 +1

z2

)M ′ = −2ξξ′

m20

, (3.14)

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JHEP03(2018)177

where we used p = m′ and

P =q

m=

ξ

m0(3.15)

from eqs. (3.10) and (3.12). Taking into account that M → 0, ξ → q0 at z =∞, eq. (3.14)

is integrated as

M ′

z−M =

q20 − ξ2

m20

. (3.16)

If there is a neck which is satisfying the conditions (3.2) and (3.11), M must have

a minimum Mm at z = zm (0 < zm < ∞).1 The lower bound of the minimum Mm is

determined by eq. (3.16) as

Mm ≥ −q2

0

m20

= −P 2|z→∞ (3.17)

where the equality is satisfied only when ξ(zm) = 0. This implies that the minimal horizon

radius RH around the neck is determined by the asymptotic value of the chemical potential

P given by eq. (3.15).

There are infinite degrees of freedom to choose a function M satisfying eq. (3.16), the

neck condition (3.2), and the lower bound (3.17). Once we choose a function M satisfying

these conditions (3.2) and (3.17), ξ and P are determined from eqs. (3.16) and (3.15).2 For

example, let us choose a Gaussian like function M :

M = −Bz4

a4e−

(z−a)2

b2 , B > 0 (3.18)

to satisfy the asymptotic boundary condition (3.11), where a, b, and B are some positive

constants. Here, we set M sufficiently rapidly approaches zero at the origin of spherical

symmetry, z = 0 to avoid a singularity. The minimum takes at

zm =a+√a2 + 8b2

2' a+

2b2

a(3.19)

in the limit a b. To satisfy the lower bound (3.17), we choose the parameter B so that

B = B0q2

0

m20

(1 +

2b2

a2

)−2

' B0q2

0

m20

, (3.20)

where B0 is a positive constant satisfying B0 < 1. Since the horizon is determined by

eq. (3.5), the cross-sectional area S defined in eq. (3.2) becomes

S′ = S(z)(nz

+M ′). (3.21)

1If there was no minimum, M would be monotonically decreasing function satisfying M > 0 for z ∈[0, ∞). This contradicts m(z0) < m0.

2Normally given P , the chemical potential on the boundary, the bulk profile M is determined. Here we

are solving this in a opposite way; given M , the bulk profile, we determine the boundary chemical potential

profile P which realizes this bulk profile M .

– 7 –

JHEP03(2018)177

26 28 30 32 34 36 38z

8

6

4

2

0

M

Figure 1. The plot of M for various values of b = 2.8 (blue, solid), 3.3 (dashed green), 3.7 (dotted

red), and 5.3 (dotdashed, brown) in the case a = n = 30, m0 = 1, q0 = 3, and B0 = 0.98.

Note that the expansion (2.4) is valid when M ′ = O(1), therefore setting a = na0 (a0 > 0),

one obtains

M ′|z=a−b ' −2B

be−1 +O

(1

n

). (3.22)

This implies that S must have a minimum around z = a if

2a0B

be−1 > 1. (3.23)

Figure 1 and figure 2 show the plots of M and S near the minimum for various values

of b in the case a = n = 30, m0 = 1, q0 = 3, and B0 = 0.98. The cross-sectional area

S monotonically increases before reaching the maximum, and then decreases toward the

minimum. This implies that the horizon behaves as a spherical black hole in the region

0 ≤ z < a, and it is connected to an asymptotic planar horizon z a through a neck

around z = a. To satisfy the condition M ′ = O(1), a must increase as n increases. So,

the position of the neck goes away from the center, z = 0, and the neck connects a large

spherical black hole with an asymptotic planar horizon, as n increases. Note that S′/S

increases with the magnitude O(n) before reaching z = O(n), and then decrease with the

magnitude O(1) at z = O(n). This implies that the mushroom shape is extremely flattened.

As shown in figure 2, a plateau region appears for each value of b, corresponding

to the neck in the black mushroom solution. This region spreads as b increases, and the

spherical black hole portion tends to disappear. These facts imply that the black mushroom

solution locally approaches a black string solution with translational symmetry along z as

b increases. As shown in figure 3, the chemical potential P possesses a precipitous valley

near the plateau region. As the external electric field E is given by P ′, the black string

portion is supported by the strong electric field.

4 Stability analysis

As seen in the previous section, we showed that there is a black mushroom solution in which

a small spherical black hole is connected to the asymptotic planar black brane through a

– 8 –

JHEP03(2018)177

24 26 28 30 32 34z0

5

10

15

20

25

30

S

Figure 2. The plot of S (normalized by S(n)) for the same values of b as figure 1 in the case

a = n = 30, m0 = 1, q0 = 3, and B0 = 0.98.

24 26 28 30 32 34 36 38z0

2

4

6

8

Figure 3. The plot of Γ = P 2 for the same values of b as figure 1 in the case a = n = 30, m0 = 1,

q0 = 3, and B0 = 0.98.

neck that resembles a black string solution. In this section, we argue the stability of the

black mushroom solution from the perspective of thermodynamics, as well as that of the

dynamical stability with respect to linear perturbations.

4.1 Stability analysis: thermodynamical argument

Gubser and Mitra have conjectured that the Gregory-Laflamme instability for black branes

with a non-compact translational symmetry occurs if and only if they are locally thermody-

namically unstable [15, 16]. This claim was proven [17]. This implies that if a black-string-

shape neck has a translationally invariant portion larger than the threshold wavelength

λc beyond which any longer wavelength perturbations are unstable, it tends to break up

under the evolution.

As shown in refs. [18, 21, 22], higher dimensional black string solution with translational

symmetry suffers from a Gregory-Laflamme instability for short wavelength perturbations.

The threshold wavelength λc is approximately given by

λc ∼ S(zm)1/(n−2) ∼ zm√n. (4.1)

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JHEP03(2018)177

Here, to derive the second approximation, we used the fact that the surface area of unit

n− 2-dimensional sphere C0 is given by C0 ∼ n−n/2 [7]. In the zm ∼ n = 30, b = 5.3 case

in the previous section, λc ∼ 5.4, which is comparable to the length of the neck, ∼ 6 (recall

that r ' 1), as seen in figure 2. So, one would expect that the neck with a translationally

invariant portion larger than λc would be unstable against Gregory-Laflamme instability

unless it is thermodynamically stable.

The temperature T for the black mushroom solution corresponds to the surface gravity

κ in eq. (3.6). As z ∼ zm ∼ n 1 in the neck, the third term proportional to p becomes

irrelevant. Then, κ is determined by the local charge q and mass parameters m on the

neck. Since κ increases as the mass increases for a fixed charge, it should be thermody-

namically stable, implying that the neck should also be dynamically stable, according to

the conjecture.

Note that the fact that the existence of the neck forces the specific heat positive is

independent of the form of M . Given M , eq. (3.21) is generic and in order to have a neck

part, we have to have zm = O(n), since M ′ = O(1). Then, from eq. (3.6), terms with

p/zm becomes O(

1n

)and we always have a positive specific heat. These suggests that in

the large D, the neck part of the mushroom solution is always stable dynamically.

4.2 Stability analysis: linear perturbation

We consider linear perturbation of the black mushroom solution satisfying eqs. (3.4). Here,

we address the issue whether the linear perturbation has an unstable mode without time

dependent external force P . So, we impose the condition

δP (t, z) = 0. (4.2)

Linearizing the evolution equations (2.12), (2.13), and (2.14), we obtain the equations for

perturbation as

˙δm+δp

z− δm′

z= 0 , (4.3)

˙δp+2p

mzδp− p2

m2zδm− 2P ′δq + δm′ −

(δp

z

)′= 0 , (4.4)

˙δq − δq′

z+δm

zP ′ +

q

zmδp+

p

zmδq − pq

zm2δm = 0 , (4.5)

where a dot and prime denote the derivative with respect to t and z, respectively. Plugging

δp = δm′ − zδm obtained from eq. (4.3) into eqs. (4.4) and (4.5), we have

zδm′′ − 2z2 ˙δm′ −(

1 + z2 + 2zp

m

)δm′

+z3 ¨δm+ 2z2 p

m˙δm+ z

p2

m2δm+ 2z2P ′δq = 0 , (4.6)

δq′ − z ˙δq − p

mδq =

q

m(δm′ − z ˙δm) +

(P ′ − pq

m2

)δm . (4.7)

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JHEP03(2018)177

Note that when p = 0 = P , the above set of equations reduce to the corresponding

perturbation equations for the large D limit of the Schwarzschild-AdS black brane solution,

which should be stable as it has a positive specific heat.

It is immediate to see from eq. (4.6) that near the center z = 0, the general solution

of δm behaves in a regular manner as

δm ' C1 + C2z2 , (4.8)

with C1, C2 being some constants independent of the values of p and P . Choosing C1 and

C2 corresponds to specifying a particular boundary condition at the center: for instance,

C1 = 0 corresponds to the Dirichlet boundary condition. Actually, which choice of the

boundary condition we would take is not relevant to the rest of our argument, and thus we

leave these constants unspecified.

It also turns out that eqs. (4.6) and (4.7) form a parabolic system. To see that, let us

change the coordinates (t, z) into (u := −t, v := 2t+ z2) so that the above two equations

are expressed as(∂2u − 2∂v −

2

z

p

m∂u +

1

z2

p2

m2

)δm+ 4z∂vPδq = 0 , (4.9)(z∂u −

p

m

)δq =

q

m

(z∂u −

p

m

)δm+ 2z∂vPδm , (4.10)

with z viewed as the function of (u, v).

Recalling the conditions (3.11) at z →∞ and also noting p = m′ → 0, we find eq. (4.10)

to become

∂uδq 'q0

m0∂uδm , (4.11)

and thus we have δq ' (q0/m0)δm. Eq. (4.9) then asymptotically takes the form of thermal

diffusion equation:

(∂2u − 2∂v)δm ' 0 . (4.12)

We naturally impose the following regularity conditions at large z:

limz→∞

δm = 0 . (4.13)

Provided the separation of variable, the above equation can be immediately solved as

δm =∑λ

a(λ)e−λ2v cos(

√2λu+ θλ)

=∑λ

a(λ)e−λ2(2t+z2) cos(

√2λt− θλ) . (4.14)

Here λ must be either a real or a pure imaginary number in the following reason. If λ

is a complex number, then the above solution could contain an unstable mode. However

if such an unstable mode is allowed, it would imply that the Schwarzschild-AdS black

– 11 –

JHEP03(2018)177

Figure 4. The construction of the “dumbbell” like black hole solution from the spherically sym-

metric black hole.

brane (p = 0 = P ) itself would admit an unstable perturbation as we have the same

expression (4.14) for the perturbations and the same boundary conditions (4.8) and (4.13)

for the case of the Schwarzschild-AdS black brane, which is however thought to be stable

from the thermodynamic perspective. Now suppose λ is pure imaginary. Then δm is

non-normalizable on t = const. surface, hence is not a physically acceptable perturbation.

Therefore, λ must be a real number, for which the perturbation solution (4.14) exhibits

no instability. It is thus plausible to argue that our black mushroom should be stable

under type of the perturbations considered above. This argument is also consistent with

the speculation that any black string portion of the neck should be stable according to

the Gubser-Mitra conjecture [15, 16], as the portion has always positive specific heat. To

fully justify this stability argument, we however need a thorough study of the dynamical

perturbations, which is the near future task.

5 Spherical black hole case

In this section, we pay close attention to the polarized AdS black hole with a spherical

horizon. If such an AdS black hole is highly squashed by external electric field, “dumbbell”

type black hole with a neck connecting two spheres appears (see figure 4). Then, as

discussed in the previous sections, Gregory-Laflamme instability would occur unless the

black string portion becomes thermodynamically stable.

One might ask whether such a polarized AdS black hole with a spherical horizon is

unstable or not, because small AdS black holes are thermodynamically unstable. In this

section, we investigate properties of such a polarized AdS black hole at large D by analyzing

1 + 1-dimensional effective equations as follows.

5.1 Derivation of effective equations

We make the metric ansatz as

ds2 = −Adt2 + 2ut dtdr − 2Cz dtdz +Gzzdz2 + r2 sin2 z dΩ2

n−2 , (5.1)

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JHEP03(2018)177

where z is the angular coordinate in the range 0 ≤ z ≤ π. As in the brane case, under the

condition (2.3), the metric is expanded as

A(r, t, z) =r2

L2

(1− m(t, z)

rn

)+ 1 +

1

n

(Q(t, z)

r2n−2+ a1(r, t, z)

)+O(n−2),

Cz(r, t, z) =p(t, z)

nrn+O(n−2),

ut(r, t, z) = 1 +βt(r, t, z)

n+O(n−2),

Gzz = r2 +H(r, t, z)

n+O(n−2),

Q = L2q2(t, z) . (5.2)

It is easily checked that the metric (5.2) and the gauge fields (2.5), (2.7) are the leading

order solutions for the spherical case. At next to leading order, we can set βt = H = 0 as

in the brane case, and we find the solution for a1 as

a1 ' −L2p cos z lnR

R sin z. (5.3)

Substituting eq. (5.3) into the Einstein equations (2.1), we obtain evolution equations for

q, m, and p as

∂q

∂t− cos z

sin z

∂q

∂z− (RHL

2 −m) cos z

sin z

∂P

∂z+

(L2 + 1) cos z

m sin zpq = 0 , (5.4)

∂m

∂t+

((1 + L2)p− ∂m

∂z

)cos z

sin z= 0 , (5.5)

∂p

∂t+

(1 + L2)p2 cos z

m sin z− 2q

∂P

∂z+

1

L2

∂m

∂z− 2p− ∂

∂z

(p cos z

sin z

)= 0 , (5.6)

where RH is the value of R at the horizon determined by A = 0.

5.2 Properties of static solutions

Here, we investigate the properties of the static spherical black hole solutions. Assuming

that q, m, and p depend on the variable z only, the static equations are reduced from

eqs. (5.4), (5.5), and (5.6) as

m′ = (1 + L2)p ,

q′ =m

1 + L2P ′ +

1 + L2

mpq ,

(p cot z)′ =m′

L2+

(1 + L2)p2

mcot z − 2qP ′ − 2p . (5.7)

The value of R at the horizon, RH , is determined by A = 0 in eq. (5.2) as

RH =m

1 + L2− 1

n

[− 2mL2

(1 + L2)2ln

(m

1 + L2

)+L2q2

m+

ma1

(1 + L2)2

]. (5.8)

– 13 –

JHEP03(2018)177

Up to O(1), the temperature T is evaluated at the value of surface gravity on the horizon,

κ =1

2A,r =

(1 + L2)n

2L2+

1− L2

2L2ln

m

1 + L2− (1 + L2)2q2

2m2− 1 +

1

2L2− (1 + L2)p cos z

2m sin z.

(5.9)

It is easily checked that κ is constant along the horizon by showing κ,z = 0 by the static

equations (5.7), as in the black mushroom case.

Now, we consider the “dumbbell” type static spherical black hole solutions in which

the equatorial plane is squashed by the external electric field. For simplicity, we assume

that the solution is symmetric with respect to the equatorial plane. This means that

p|z=π2

= m′|z=π2

= 0. (5.10)

We also assume that M sufficiently quickly approaches zero at the north pole (also south

pole) as

M = cz2+ε, ε > 0, (5.11)

as in the black mushroom case.

The total mass and the charge M and Q are determined by the mass and charge

density m and q as

M∼∫ π

0m(z) sinn−2 z dz, Q ∼

∫ π

0q(z) sinn−2 z dz. (5.12)

This implies thatM andQ are dominated by the values of m|z=π/2 := me and q|z=π/2 := qe,

respectively in the large n limit since sinn−2 z becomes zero except z = π/2 in the limit.

At the equatorial plane, by eq. (5.10), κ is rewritten by me and qe as

κ =(1 + L2)n

2L2+

1− L2

2L2ln

me

1 + L2− (1 + L2)2q2

e

2m2e

− 1 +1

2L2. (5.13)

Therefore, the condition for the negative specific heat becomes

L > 1 and 2(1 + L2)2q2e <

L2 − 1

L2m2e . (5.14)

Let us define M and ξ by eq. (3.13). Here, m0 and q0 are defined by the values of

north pole, respectively:

m0 := m|z=0, q0 := q|z=0. (5.15)

Eliminating p from eqs. (5.7) and integrating the second equation of (5.7), we find

q0 =Pm0

1 + L2, (5.16)

where we used the regularity condition At = 0 on the horizon. Eliminating P from the

third equation in (5.7) by eq. (5.16), we obtain

M ′′ cot z +

(1− 1

L2− 1

sin2 z

)M ′ = −2(1 + L2)2ξξ′

m20

. (5.17)

– 14 –

JHEP03(2018)177

The first integration yields

M ′ cot z +

(1− 1

L2

)M =

(1 + L2)2

m20

(q20 − ξ2)

= (1 + L2)2

(q2

0

m20

− q2

m2

), (5.18)

where we used ξ2/m20 = q2/m2 and the boundary condition (5.11). Therefore, we obtain

M |z=π2

=L2(1 + L2)2

(L2 − 1)

(q2

0

m20

− q2e

m2e

)> −L

2(1 + L2)2

(L2 − 1)

q2e

m2e

> − 1

2L2> −1

2(5.19)

under the condition (5.14). This is the lower bound of M at the equatorial plane, which

means that the equatorial plane cannot be highly squashed, keeping the negative specific

heat. In other words, highly squashed equatorial black dumbbell is possible to construct

but its specific heat is always positive. According to the Gubser-Mitra conjecture [15], this

indicates that Gregory-Laflamme instability does not occur in the “dumbbell” type static

spherical black hole solutions.

5.3 Linear perturbations

We consider linear perturbation of the static black hole solutions satisfying eqs. (5.7). As

in the black brane case, we assume that the perturbation of P is zero. Then, linearizing

the evolution equations (5.4), (5.5), and (5.6), we obtain

˙δq −(δq′ − P ′δm

1 + L2+

(1 + L2)pq

m2δm

)cot z +

1 + L2

m(qδp+ pδq) cot z = 0 , (5.20)

˙δm+ (1 + L2)δp− δm′ cot z = 0 , (5.21)

˙δp+ (1 + L2)

(2pδp

m− p2

m2δm

)cot z − 2P ′δq +

δm′

L2− 2δp− (δp cot z)′ = 0 . (5.22)

From the regularity on the equatorial plane z = π/2, the following boundary conditions

are derived:

˙δq|z=π2

= ˙δm|z=π2

= 0 . (5.23)

Note that this is consistent with the mass and charge conservation law, i.e., M and Qdefined in eq. (5.12) are constant during the time evolution in the large n limit.

Eliminating δp by using eq. (5.21), we obtain two equations for δm and δq as follows:

˙δq − cot zδq′ + (1 + L2)p

mcot zδq − cot z

[(1 + L2)

pq

m2− P ′

1 + L2

]δm

+(1 + L2)q

mcot zδm′ − q

m˙δm = 0 , (5.24)

2 ˙δm′ − 1

cot z¨δm− cot zδm′′ +

[cot2 z +

1

L2

]δm′ +

2

cot z˙δm

+2(1 + L2)p

m(cot zδm′ − ˙δm)− (1 + L2)2 cot z

p2

m2δm− 2(1 + L2)P ′δq = 0 . (5.25)

The stability analysis from now on parallels what we have done below eqs. (4.6) and (4.7)

for our black branes. We can make the same plausible argument for our black dumbbell,

– 15 –

JHEP03(2018)177

agreeing with the thermodynamical argument that the Gregory-Laflamme type instability

does not occur in the squashed black holes.

6 Summary and discussions

In this paper, we have first derived a new set of effective equations (2.12)–(2.14), de-

scribing the dynamics of the polarized black branes (holes) to leading order in the 1/D

expansion and using these, we have tested cosmic censorship conjecture in polarized AdS

black brane (hole) solutions at large D dimensions. As expected in the four-dimensional

analysis [4], we found a black mushroom solution where a black hole is connected with an

asymptotic planar black brane through a black-string-shape neck under the localized chem-

ical potential. Contrary to our first naive expectation, the black-string-shape neck part is

thermodynamically stable. This indicates that the localized string cannot be pinched off

dynamically according to the Gubser-Mitra conjecture [15, 16]. We have extended the

analysis to the AdS black hole case and found that highly squashed black hole is also

dynamically and thermodynamically stable. These facts imply that the cosmic censor-

ship is not violated in such polarized AdS black brane (hole) solutions at large D by the

Gregory-Laflamme instability [5].

For simplicity, we have treated the gauge field as a probe approximation in the sense

that the horizon geometry at leading order is neutral black brane (hole) solutions. In other

words, the horizon is embedded at the fixed bulk radial coordinate in AdS spacetime in the

leading order. To take into account the gauge field at leading order, we must construct a

charged polarized black brane (hole) solutions at leading order so that the horizon is located

over different radial region. It is interesting to test the cosmic censorship conjecture in that

case. This will be investigated in the near future.

Acknowledgments

We would like to thank Kentaro Tanabe and Norihiro Tanahashi for discussions in the

early stage of the project. We would especially like to thank Kentaro Tanabe for sharing

his unpublished notes [23] with us in the early stage of the project, and Roberto Emparan

for valuable comments on various aspects of our results. We would also like to thank

Gary T. Horowitz and Ryotaku Suzuki for useful comments on the manuscript. This work

was supported in part by JSPS KAKENHI Grant Number 25800143 (NI), 15K05092 (AI),

17K05451 (KM).

A Formula for curvature decomposition

D-dimensional Ricci curvature on the metric ansatz (2.2) is decomposed into Ricci curva-

ture and the Christoffel symbol on the three-dimensional spacetime (t, r, z) as

Rrr = R(3)rr + (n− 2)

(Γrrrr

+Γzrrz

),

Rrz = R(3)rz + (n− 2)

(Γzzrz

+Γrrzr− 1

rz

),

– 16 –

JHEP03(2018)177

Rrt = R(3)rt + (n− 2)

(Γzrtz

+Γrrtr

),

Rtt = R(3)tt + (n− 2)

(Γzttz

+Γrttr

),

Rzz = R(3)zz + (n− 2)

(Γzzzz

+Γrzzr

),

Rtz = R(3)tz + (n− 2)

(Γztzz

+Γrtzr

). (A.1)

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

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